
where \(S^{(q)}(t\mid \cdot ) = 1\)  for all \(t<0\)  and \(S^{(q)}(t\mid \gamma ^{(q-1)} = 1) = 1(t <0)\) , for \(q=Q-1, Q-2, ..., 1.\)  In other words, assuming no censoring events, the survival probability \(S^{(q)}\)  of the remaining life \(L^{(q)}\)  at stage \(q\)  can be obtained by augmenting the survival probability of the remaining life \(L^{(q+1)}\)  of the next stage by the current stage length \(V^{(q)}\)  and then marginalizing over all events that occur later than the beginning of stage \(q\) . Under the covariate-independent censoring assumption, the contribution of the censored cases is incorporated into the equation by redistributing the probability to the right [1]}:
\(& S^{(q)}(t\mid H^{(q)}, A^{(q)})\nonumber \\= & \int \bigg [\delta ^{(q)}S^{(q+1)}(t - X^{(q)}\mid H^{(q+1)}, A^{(q+1)}) \nonumber \\& ~ ~ ~ ~ ~ +(1-\delta ^{(q)})\Big \lbrace 1(X^{(q)}> t) + 1(X^{(q)}\le t) \frac{S^{(q)}(t\mid H^{(q+1)}, A^{(q+1)})}{S^{(q)}(X^{(q)}\mid H^{(q+1)}, A^{(q+1)})}\Big \rbrace  \bigg ]\nonumber \\&~ ~ ~ \times dP(X^{(q)}, H^{(q+1)}, A^{(q+1)}, \delta ^{(q)}\mid H^{(q)}, A^{(q)}). \) 
